Xenharmonic Wiki

User:Ganaram inukshuk/5L 2s: Difference between revisions

← Older edit
VisualWikitext
Ganaram inukshuk (talk | contribs)
autopatrolled
4,810 edits
ArrowHead294 (talk | contribs)
14,512 edits
mNo edit summary
 
(43 intermediate revisions by one other user not shown)
Line 1: Line 1:
{{Infobox MOS|Tuning=5L 2s}}
{{Infobox MOS|Tuning=5L 2s|debug=1}}
:''This is a test page. For the main page, see [[5L 2s]].''
:''This is a test page. For the main page, see [[5L 2s]].''
{{MOS intro|Scale Signature=5L 2s}}
{{MOS intro|Scale Signature=5L 2s}}
==Name==
Among the most well-known forms of this scale are the diatonic scale of [[12edo]], the Pythagorean diatonic scale, and scales produced by meantone systems.
TAMNAMS suggests the name '''diatonic''' for this scale, referring to the use of this term to refer to a scale with 5 whole steps and 2 small steps.
== Name==
[[TAMNAMS]] suggests the temperament-agnostic name '''diatonic''' for this scale, which commonly refers to a scale with 5 whole steps and 2 small steps. Under TAMNAMS and for all scale pattern pages on the wiki, '''the term ''diatonic'' exclusively refers to 5L 2s'''.


===On the term ''diatonic''===
The term ''diatonic'' may also refer to scales produced using [[Tetrachord|tetrachords]], [[just intonation]], or in general have more than one size of whole tone. Such scales, such as [[Zarlino]], [[blackdye]] and [[diasem]], are specifically called ''[[Detempering|detempered]] diatonic scales'' (for an RTT-based philosophy) or ''deregularized diatonic scales'' (for an RTT-agnostic philosophy). The terms ''diatonic-like'' or ''diatonic-based'' may also be used to refer such scales, depending on what's contextually the most appropriate.
Under TAMNAMS and for all scale pattern pages on the wiki, '''the term ''diatonic'' exclusively refers to 5L 2s'''. Other diatonic-based scales, such as [[Zarlino]], [[blackdye]] and [[diasem]], are called ''[[Detempering|detempered]] diatonic scales'' (for an RTT-based philosophy) or ''deregularized diatonic scales'' (for an RTT-agnostic philosophy). The terms ''diatonic-like'' or ''diatonic-based'' may also be used to refer to diatonic-based scales, depending on what's contextually the most appropriate.


==Notation==
==Intervals==
:''This article assumes [[TAMNAMS]] for naming step ratios, mossteps, and mosdegrees.''
Names for this scale's intervals (mossteps) and scale degrees (mosdegrees) are based on the number of large and small steps from the root, starting at 0 (the 0-mosstep and 0-mosdegree) for the unison.


===Intervals===
Except for the unison and octave, all [[Interval class|interval classes]] have two [[Interval variety|varieties]] or sizes, denoted using the terms ''major'' and ''minor'' for the large and small sizes, respectively. The exception to this rule are the generators, which use the terms ''augmented'', ''perfect'', and ''diminished'' instead.
Intervals are identical to that of standard notation. As such, the usual interval qualities of major/minor and augmented/perfect/diminished apply here.
{| class="wikitable"
{| class="wikitable"
! rowspan="2" |Interval class
|+5L 2s interval varieties
! colspan="2" |Large variety
!Interval class
! colspan="2" |Small variety
!Specific intervals
!Size (in ascending order)
|-
|-
!Size
|'''0-diastep'''
!Quality
|'''Perfect 0-diastep (unison)'''
!Size
|0
!Quality
|-
|-
|'''1st (unison)'''
| rowspan="2" |1-diastep
|0
|Minor 1-diastep
|Perfect
|s
|0
|Perfect
|-
|-
|2nd
|Major 1-diastep
|L
|L
|Major
|s
|Minor
|-
|-
|3rd
| rowspan="2" |2-diastep
|Minor 2-diastep
|L + s
|-
|Major 2-diastep
|2L
|2L
|Major
|L + s
|Minor
|-
|-
|4th
| rowspan="2" |'''3-diastep'''
|'''Perfect 3-diastep'''
|2L + s
|-
|Augmented 3-diastep
|3L
|3L
|Augmented
|2L + 1s
|Perfect
|-
|-
|5th
| rowspan="2" |'''4-diastep'''
|3L + 1s
|Diminished 4-diastep
|Perfect
|2L + 2s
|2L + 2s
|Diminished
|-
|-
|6th
|'''Perfect 4-diastep'''
|4L + 1s
|3L + s
|Major
|-
| rowspan="2" |5-diastep
|Minor 5-diastep
|3L + 2s
|3L + 2s
|Minor
|-
|-
|7th
|Major 5-diastep
|5L + 1s
|4L + s
|Major
|-
| rowspan="2" |6-diastep
|Minor 6-diastep
|4L + 2s
|4L + 2s
|Minor
|-
|-
|'''8th (octave)'''
|Major 6-diastep
|5L + s
|-
|'''7-diastep (octave)'''
|'''Perfect 7-diastep (octave)'''
|5L + 2s
|5L + 2s
|Perfect
|5L + 2s
|Perfect
|}
|}
A 7-note scale using these intervals will typically use scale degrees that represents one size from each interval class, with the true MOS upholding the step pattern of LLLsLLs, or some rotation thereof. MODMOS scales may be formed this way without upholding the step pattern, thereby creating a non-MOS pattern such as LLLLsLs, or may include alterations that exceed the two varieties typical of a MOS scale.


===Note names===
==Notation==
Note names are identical to that of standard notation. Thus, the basic gamut for 5L 2s is the following:
:''See [[5L 2s/Notation]]''
 
{{MOS gamut|Scale Signature=5L 2s}}
==Theory==
==Theory==


===5L 2s as a moment-of-symmetry scale===
===Introduction to step sizes===
The familiar interpretation of whole and half steps, commonly written as WWHWWWH for the major scale, has step sizes of 2 (whole step) and 1 (small step), producing [[12edo]]. Viewing 5L 2s as a moment-of-symmetry scale involves generalizing the step pattern of 5 whole steps and 2 whole steps as being different step sizes. As such, the generalized form LLsLLLs is used, as most step ratios have step sizes that cannot be interpreted as being "whole" or "half" steps.
:''Main article: [[Scale tree]] and [[TAMNAMS#Step ratio spectrum]]''
The familiar pattern of 5 whole steps and 2 half steps, commonly written as WWHWWWH for the major scale, has step sizes of 2 (whole step) and 1 (half step), producing [[12edo]]. This can be generalized into the form LLsLLLs, with whole-number sizes for the large steps and small steps, denoted as "L" and "s" respectively.


====Substituting step sizes====
Different edos are produced by using different ratios of step sizes. A few examples are shown below.
Different edos are produced by substituting different step sizes. A few examples are shown below.
{| class="wikitable"
{| class="wikitable"
|+
|+
Line 87: Line 85:
!Step pattern
!Step pattern
!EDO
!EDO
! Selected multiples
|-
|1:1
|1 1 1 1 1 1 1
|[[7edo]]
|[[14edo]], [[21edo]], etc.
|-
|-
|4:3
|4:3
|4 4 3 4 4 4 3
| 4 4 3 4 4 4 3
|[[26edo]]
|[[26edo]]
|
|-
|-
|3:2
|3:2
|3 3 2 3 3 3 2
|3 3 2 3 3 3 2
|[[19edo]]
|[[19edo]]
|[[38edo]]
|-
|-
|5:3
| 5:3
|5 5 3 5 5 5 3
| 5 5 3 5 5 5 3
|[[31edo]]
|[[31edo]]
|
|-
|-
|2:1
| 2:1
|2 2 1 2 2 2 1
|2 2 1 2 2 2 1
|[[12edo]] (standard tuning)
|[[12edo]] (standard tuning)
|[[24edo]], [[36edo]], etc.
|-
|-
|5:2
|5:2
|5 5 2 5 5 5 2
| 5 5 2 5 5 5 2
|[[29edo]]
|[[29edo]]
|
|-
|-
|3:1
|3:1
|3 3 1 3 3 3 1
|3 3 1 3 3 3 1
|[[17edo]]
|[[17edo]]
|[[34edo]]
|-
|-
|4:1
|4:1
|4 4 1 4 4 4 1
|4 4 1 4 4 4 1
|[[22edo]]
|[[22edo]]
|
|-
|1:0
|1 1 0 1 1 1 0
|[[5edo]]
|[[10edo]], [[15edo]], etc.
|}
|}
Edos that are multiples of 12, such as [[24edo]] or [[72edo]], are reached by using step ratios that are not simplified, such as 4:2 for 24edo and 12:6 for 72edo. The step sizes may be called whole and half in this case.
Edos that are multiples of the examples above can be reached by entering non-simplified step ratios. For example, edos that are multiples of 12 are reached by using larger values whose ratio simplifies to 2:1, such as 4:2 for [[24edo]].


====Expanding the tuning spectrum====
All step ratios lie on a spectrum from 1:1 to 1:0, referred to on the wiki as a scale tree. The step ratios 1:1 and 1:0 represent the limits for valid step ratios. A step ratio that approaches 1:1, where the large and small step are equal to one another, approaches [[7edo]], and a step ratio that approaches 1:0, where the small step "collapses" to zero, approaches [[5edo]].
The tuning spectrum shown in the previous section is produced by starting with the step ratios 1:1 and 1:0 and repeatedly finding the [[Mediant|mediants]] between adjacent ratios. The first three iterations are shown below.
{{SB tree|Depth=1}}
{{SB tree|Depth=2}}
{{SB tree|Depth=3}}
Larger edos, such as [[53edo]] or [[311edo]], can be reached by repeatedly expanding the tuning spectrum. The section tuning spectrum contains a much larger tuning spectrum.


The step ratios 1:1 and 1:0 represent the extremes of the tuning spectrum. A step ratio that approaches 1:1, where the large and small step are equal to one another, approaches [[7edo]], and a step ratio that approaches 1:0, where the size of the small step approaches 0 relative to the size of the large step, approaches [[5edo]].
TAMNAMS has names for regions of this spectrum based on whether they are "soft" (between 1:1 and 2:1) or "hard" (between 2:1 and 1:0).


===Temperament interpretations===
===Temperament interpretations===
:''Main article: [[5L 2s/Temperaments]]''
5L 2s has several rank-2 temperament interpretations, such as:
*[[Meantone]], with generators around 696.2¢. This includes:
**[[Flattone]], with generators around 693.7¢.
*[[Schismic]], with generators around 702¢.
*[[Parapyth]], with generators around 704.7¢.
*[[Archy]], with generators around 709.3¢. This includes:
**Supra, with generators around 707.2¢
**Superpyth, with generators around 710.3¢
**Ultrapyth, with generators around 713.7¢.
==Tuning ranges==
==Tuning ranges==
===Simple tunings===
[[17edo]] and [[19edo]] are the smallest edos that offer a greater variety of pitches than 12edo. Note that any enharmonic equivalences that 12edo has no longer hold for either 17edo or 19edo, as shown in the table below.
===Parasoft tunings ===
:''Main article: [[Flattone]]''
Parasoft tunings (4:3 to 3:2) correspond to flattone temperaments, characterized by flattened perfect 5ths ([[3/2]], flat of 702¢) to produce major 3rds that are flatter than [[5/4]] (386¢).
Edos include [[19edo]], [[26edo]], [[45edo]], and [[64edo]].
===Hyposoft tunings===
:''Main article: [[Meantone]]''
Hyposoft tunings (3:2 to 2:1) correspond to meantone temperaments, characterized by flattened perfect 5ths (flat of 702¢) to produce diatonic major 3rds that approximate 5/4 (386¢).
Edos include [[19edo]], [[31edo]], [[43edo]], and [[50edo]].
===Hypohard tunings===
:''Main article: [[Pythagorean tuning]] and [[Schismatic family#Schismatic aka Helmholtz|schismatic temperament]]''
The range of hypohard tunings can be divided into a minihard range (2:1 to 5:2) and quasihard range (5:2 to 3:1).
====Minihard tunings====
Minihard tunings correspond to Pythagorean tuning and schismatic temperament, characterized by having a perfect 5th that is as close to just (701.96¢) as possible, resulting in a major 3rd of [[81/64]] (407¢).
Edos include [[41edo]] and [[53edo]].
==== Quasihard tunings====
Quasihard tunings correspond to "neogothic" or "parapyth" systems whose perfect 5th is slightly sharper than just, resulting in major 3rds that are sharper than 81/64 and minor 3rds that are slightly flat of [[32/27]] (294¢).
Edos include [[17edo]], [[29edo]], and [[46edo]]. 17edo is considered to be on the sharper end of the neogothic spectrum, with a major 3rd that is more discordant than flatter neogothic tunings.
===Parahard and ultrahard tunings===
:''Main article: [[Archy]]''
Parahard (3:1 to 4:1) and ultrahard tunings (4:1 to 1:0) correspond to archy systems, with perfect 5ths that are significantly sharper than than 702¢.
Edos include [[17edo]], [[22edo]], [[27edo]], and [[32edo]], among others.
==Modes==
==Modes==
Diatonic modes have standard names from classical music theory:
Diatonic modes have standard names from classical music theory:
{{MOS modes|Scale Signature=5L 2s}}
Each mode has the following scale degrees, reached by raising or lowering certain naturals by a chroma.
{| class="wikitable"
|+TODO: Make this into a template
! colspan="2" |Mode
! colspan="8" |Scale degree (on C)
|-
!UDP
!Step pattern
!1st
!2nd
!3rd
!4th
!5th
! 6th
!7th
!8th
|-
|<nowiki>6|0</nowiki>
|LLLsLLs
|Perfect (C)
| Major (D)
|Major (E)
|Augmented (F#)
|Perfect (G)
|Major (A)
|Major (B)
|Perfect (C)
|-
|<nowiki>5|1</nowiki>
|LLsLLLs
|Perfect (C)
|Major (D)
|Major (E)
|Perfect (F)
|Perfect (G)
|Major (A)
|Major (B)
| Perfect (C)
|-
|<nowiki>4|2</nowiki>
|LLsLLsL
|Perfect (C)
|Major (D)
|Major (E)
|Perfect (F)
|Perfect (G)
|Major (A)
|Minor (Bb)
|Perfect (C)
|-
|<nowiki>3|3</nowiki>
|LsLLLsL
| Perfect (C)
|Major (D)
|Minor (Eb)
|Perfect (F)
|Perfect (G)
|Major (A)
|Minor (Bb)
|Perfect (C)
|-
|<nowiki>2|4</nowiki>
|LsLLsLL
|Perfect (C)
|Major (D)
|Minor (Eb)
| Perfect (F)
|Perfect (G)
|Minor (Ab)
|Minor (Bb)
|Perfect (C)
|-
|<nowiki>1|5</nowiki>
|sLLLsLL
|Perfect (C)
|Minor (Db)
| Minor (Eb)
|Perfect (F)
|Perfect (G)
|Minor (Ab)
|Minor (Bb)
|Perfect (C)
|-
|<nowiki>0|6</nowiki>
|sLLsLLL
|Perfect (C)
|Minor (Db)
|Minor (Eb)
|Perfect (F)
|Diminished (Gb)
|Minor (Ab)
|Minor (Bb)
|Perfect (C)
|}
== Generator chain ==
''Explain how the scale can also be generated by stacking 6 generating intervals in any combination of up or down from the root. Also explain how this can be extended further to 11 generators to produce chromatic scales.''


{{MOS modes|Scale Signature=5L 2s}}
==Scales==
==Scales==
==Tuning spectrum==
 
===Subset and superset scales===
5L 2s has a parent scale of [[2L 3s]], a pentatonic scale, meaning 2L 3s is a subset. 5L 2s also has the two child scales, which are supersets of 5L 2s:
 
*[[7L 5s]], a chromatic scale produced using soft-of-basic step ratios.
*[[5L 7s]], a chromatic scale produced using hard-of-basic step ratios.
 
12edo contains 5L 2s as the equalized form of both 5L 7s and 7L 5s.
 
===MODMOS scales and muddles===
{{main| 5L 2s MODMOSes }} ''and [[5L 2s Muddles]]''
 
===Scala files===
*[[Meantone7]] – 19edo and 31edo tunings
*[[Nestoria7]] – 171edo tuning
*[[Pythagorean7]] – Pythagorean tuning
*[[Garibaldi7]] – 94edo tuning
*[[Cotoneum7]] – 217edo tuning
*[[Pepperoni7]] – 271edo tuning
*[[Supra7]] – 56edo tuning
*[[Archy7]] – 472edo tuning
 
==Scale tree ==
{{MOS tuning spectrum
| Scale Signature = 5L 2s
| Depth = 6
| 7/5 = [[Flattone]] is in this region
| 21/13 = [[Golden meantone]] (696.2145{{c}})
| 5/3 = [[Meantone]] is in this region
| 2/1 = (Generators smaller than this are proper)
| 9/4 = The generator closest to a just [[3/2]] for EDOs less than 200
| 16/7 = [[Garibaldi]] / [[Cassandra]]
| 21/8 = Golden neogothic (704.0956{{c}})
| 8/3 = [[Neogothic]] is in this region
| 4/1 = [[Archy]] is in this region
}}
 
==See also==
* [[Diatonic functional harmony]]
Retrieved from "https://en.xen.wiki/w/User:Ganaram_inukshuk/5L_2s"